Welcome to JA and thanks for this question. Im sorry to hear about your symptoms. I just came online to see that you had been waiting for some time so ill quickly answer this question for you.Now a dry cough of this sort can be due to a number of different conditions, but tghe most likely one here is that he could have developed an infection of the lungs. Given that there is a dry cough, the most likely infection is bronchitis which is inflammation of the airways in the lungs. This would cause an obstruction in the passage of air into and out of the lungs leading to repeated coughing. This may also cause wheezing in very severe cases but wheezing may be absent in many cases as well. This would need to be diagnosed with the help of a clinical examination and possibly even a chest xray. In most cases using simple medications should be able to control the symptoms however given your son's past diagnosis of MELAS, we need to determine which medications can be safely used.
In case of bronchitis, he may need a short course of steroids which would need to be STRICTLY monitored along with a course of antibiotics likely to be azithromycin which is safe to use in MELAS. For now you can give him OTC medications to try and control the cough, these can include drugs like antihistamines such as zyrtec, benadryl or claritin. Additionally he can also use OTC anti cough mecications like robitussin or mucinex. You can have him do steam inhalation with oil of eucalyptus in the steam which can help in this situation as well.
But for now i think you can give him the anti cough medication along with the antihistamines which can help with the cough. if the cough is so severe that he is unable to breathe properly then take him immediately to the ER. Also here is a list of medications which should be avoided or used under monitoring in his condition. It will help you whenever he is prescribed any medications.
Link for medications in MELAS
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